Pattern Formation in Bacterial Colonies with Density-Dependent Diffusion
Recent experiments have shown that patterns can emerge in bacterial colonies programmed to have a drop in diffusion when population densities (detected via a quorum sensing mechanism) are sufficiently large. We examine one partial differential equation model of this system, and construct its non-constant stationary solutions. We demonstrate analytically that these solutions are stable when the diffusion rate of bacteria is large and the diffusion rate of signalling molecules, D_h, is small. We further demonstrate that increasing D_h induces a Hopf bifurcation, resulting in a loss of stability. These results are confirmed by numerical simulations.